The total magnetic field is found by summing (\ref{eqn:H_A}) and
(\ref{eqn:H_F1})--(\ref{eqn:H_F4}).
\begin{align}
    \vec{H} &= \vec{H}_A + \vec{H}_F\label{eqn:Ht1}\\    		&=\nabla\times\vec{A}+\frac{\nabla\bigl(\nabla\cdot\vec{F}\bigr)}{j\omega\mu'}-j\omega\varepsilon'\vec{F}\label{eqn:Ht2}\\
    &=\nabla\times\vec{A}+\frac{1}{j\omega\mu'}\biggl[\nabla\bigl(\nabla\cdot\vec{F}\bigr)-(\gamma^2=-k^2)\vec{F}\biggr]\label{eqn:Ht3}\\
    &=\nabla\times\vec{A}+\frac{1}{j\omega\mu'}\biggl[\nabla\bigl(\nabla\cdot\vec{F}\bigr)-\nabla^2\vec{F}-\vec{M}_i\biggr]\label{eqn:Ht4}\\
    &=\nabla\times\vec{A}+\frac{1}{j\omega\mu'}\biggl[\nabla\times\nabla\times\vec{F}-\vec{M}_i\biggr]\label{eqn:Ht5}
\end{align}
The total electric field is found by summing (\ref{eqn:EsubF}) and
(\ref{eqn:E_A1})--(\ref{eqn:E_A4}).
\begin{align} \vec{E}&=\vec{E}_F+\vec{E}_A\label{eqn:Et1}\\
&=-\bigl(\nabla\times\vec{F}\bigr)+\frac{\nabla\bigl(\nabla\cdot\vec{A}\bigr)}{j\omega\varepsilon'}-j\omega\mu'\vec{A}\label{eqn:Et2}\\ &=-\bigl(\nabla\times\vec{F}\bigr)+\frac{1}{j\omega\varepsilon'}\biggl[\nabla\bigl(\nabla\cdot\vec{A}\bigr)-(\gamma^2=-k^2)\vec{A}\biggr]\label{eqn:Et3}\\    &=-\bigl(\nabla\times\vec{F}\bigr)+\frac{1}{j\omega\varepsilon'}\biggl[\nabla\bigl(\nabla\cdot\vec{A}\bigr)-\nabla^2\vec{A}-\vec{J}_i\biggr]\label{eqn:Et4}\\
&=-\bigl(\nabla\times\vec{F}\bigr)+\frac{1}{j\omega\varepsilon'}\biggl[\nabla\times\nabla\times\vec{A}-\vec{J}_i\biggr]\label{eqn:Et5}
\end{align}
